On almost rigid rotations. Part 2

Abstract

The dynamical properties of a fluid, occupying the space between two concentric rotating spheres, are considered, attention being focused on the case where the angular velocities of the spheres are only slightly different and the Reynolds number R of the flow is large. It is found that the flow properties differ inside and outside a cylinder [Cscr ], circumscribing the inner sphere and having its generators parallel to the axis of rotation. Outside [Cscr ] the fluid rotates as if rigid with the angular velocity of the outer sphere. Inside [Cscr ] the fluid rotates with an angular velocity intermediate to the angular velocities of the two spheres and determined by the condition that the flux of fluid into the boundary layer of the faster-rotating sphere is equal to the flux out of the boundary layer of the slower-rotating sphere at the same distance from the axis. The return of fluid is effected by a shear layer near [Cscr ] and we show that it has a complicated structure for it can be divided into three separate layers, two outer ones, of thickness $\sim R^{-\frac{2}{7}}$ and ∼R−¼, and an inner layer of thickness $\sim R^{-\frac{1}{3}}$.

Successive elimination of shear layers by a hierarchy of constraints in inviscid spherical-shell flows

“…In both cases, the azimuthal component of flow has the same (inviscid) structure u φ ∼ (r 2 i − s 2 ) −1/2 , although, under equatorial symmetry, the singularity is arguably more severe than in the equatorially antisymmetric case, since u s and u z are also discontinuous on C. Viscosity smoothes this singularity into a shear layer comprising an inner and outer layer on either side of the tangent cylinder. For both symmetries, these inner and outer layers have a thickness that appears to scale as O(E 2/7 ) and O(E 1/4 ), respectively, scalings identical to the boundary-forced case 22 and with numerical studies of inertial waves with weak viscosity. 19 Interestingly, the layer in which the shear takes its peak value depends on the symmetry class: the E S (E A ) example taking its peak value in the outer (inner) boundary layer.…”

“…Essentially, in what may be viewed as a generalisation of the Taylor-Proudman theorem, the impenetrable boundary conditions on the inner sphere are communicated only in the direction parallel to the rotation axis and this leads to singularities or discontinuities in all components of the driven flow on the tangent cylinder, C. We have set out a hierarchy of conditions, imposed on the body force f, that progressively smoothes the driven flow on C. The first condition, A 0 = 0, ensures that u s is continuous, the second, A 1 = 0, removes the singularity in u φ and if A 0 = A 1 = A 2 = 0 then all three components of the flow are continuous. This hierarchy of conditions on f is analogous to the nested sequence of boundary layers found in similar problems that retain a small viscosity, 12,22 each additional layer removing a higher order discontinuity from the flow. In the inviscid case considered, a smooth flow (i.e., infinitely differentiable) on C must be driven by a force satisfying infinitely many constraints.…”

“…A m,n (z) s m+2n e i m φ (22) and therefore (21) is guaranteed to be finite when m > 0. In the axisymmetric case, we may write…”

“…Motivated by some of the known asymptotic scalings in both the boundary-driven 22 and forcedriven 21 problems, we plot suitably rescaled solutions in the panels on the right. In both rows, the coincidence of the horizontal location of features a to f , rescaled from features a to f, supplies strong numerical evidence of inner and outer boundary layers of respective thickness E 2/7 and E 1/4 .…”

“…In a rotating, spherical-shell geometry, analytic progress dates back to the seminal work of Proudman 18 and Stewartson 22 in which they considered a low-viscosity system which was driven through a small relative difference, 1, in the angular velocity of the boundary surfaces. The primary flow driven by the motion of the boundaries is that of a simple solid body rotation, although of considerably more interest is the secondary flow of magnitude O( ) which is induced by viscous stresses at the boundaries.…”